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The spectral even cycle problem
Abstract
In this paper, we study the maximum adjacency spectral radii of graphs of large order that do not contain an even cycle of given length. For \(n›k\), let \(S_{n,k}\) be the join of a clique on \(k\) vertices with an independent set of \(n-k\) vertices and denote by \(S_{n,k}^+\) the graph obtained from \(S_{n,k}\) by adding one edge. In 2010, Nikiforov conjectured that for \(n\) large enough, the \(C_{2k+2}\)-free graph of maximum spectral radius is \(S_{n,k}^+\) and that the \(\{C_{2k+1},C_{2k+2}\}\)-free graph of maximum spectral radius is \(S_{n,k}\). We solve this two-part conjecture.
Mathematics Subject Classifications: 05C35, 05C50
Keywords: Spectral Turán number, even-cycle problem, Brualdi-Solheid problem